Fundamentals of Time and Relativity

Minkowski Product

  • The metric of Minkowski space induces the definition of an invariant inner product between vectors.
  • Like spacetime intervals, vectors in Minkowski space can be uniquely classified into timelike, spacelike and null (lightlike) vectors.

In Minkowski space one can the define a set of four base vectors { e 0 , e 1 , e 2 , e 3 } . The index 0 refers to the time coordinate x 0 , and the other three indices to the spatial coordinates x 1 ,  x 2 ,  x 3 . The base vectors are orthonormal in the sense of the Lorentz orthonormality condition (4.6) below. A reference frame is inertial if e 0 is a future-pointing timelike vector.

In terms of these basis vectors, the displacement between two points in spacetime can be expressed as Δs=Δ x μ e μ and a general vector in this 4-space can be written in the form

4.5

x= x 0 e 0 + x 1 e 1 + x 2 e 2 + x 3 e 3 = x μ e μ ,  μ=0,1,2,3

This very much looks like a vector in Euclidean space because of the plus signs in this expression. However, the minus sign is hidden in the normalization of the base vectors which is such that e 0 2 =1,  e i 2 =−1 ; see (4.6). (It is conventional to denote four-vectors in italics, and not in boldface, as is common for Euclidean vectors.)

The invariant Minkowski inner product, indicated by a dot, between any two four-vectors u and v is defined according to the rules:

4.6

e μ ⋅ e ν = g μν     e μ ⋅ e ν = δ μ ν    

4.7

u⋅v:= g μν u μ v ν = g μν u μ v ν = δ ν μ u μ v ν = u μ v μ

The 4-dimensional Kronecker delta δ μ ν =1 , if μ=ν , and zero otherwise. The above equations imply the invariant Minkowski norm squared

4.8

u 2 = ( u 0 ) 2 − ( u 1 ) 2 − ( u 2 ) 2 − ( u 3 ) 2

by which four-vectors can be classified into timelike, spacelike and null, just like spacetime intervals.

  • The Minkowski inner product has the usual properties of an inner product but one: the Minkowski inner product is not positive definite.
  • Einstein's definition of simultaneity in an inertial frame of reference corresponds to orthogonality e 0 ⋅ e i =0, i=1,2,3 of the time and space axes in Minkowski space.